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Chapter 4: Q47E (page 137)

An electron in an atom can "jump down" from a higher energy level to a lower one, then to a lower one still. The energy the atom thus loses at each jump goes to a photon. Typically, an electron might occupy a level for a nanosecond. What uncertainty in the electron's energy does this imply?

Short Answer

Expert verified

The uncertainty in the electron’s energy is ∆E≥5.3×10-23J.

Step by step solution

01

Given data

Time is given as: ∆t=10-9s.

02

Uncertainty principle

The principle states that the position and the velocity of an object cannot be measured with 100 % accuracy at the same time.

∆x·∆p≥h2

∆x= Uncertainty in the position.

∆p = Uncertainty of momentum.

h = Planck's constant. = 1.05x10-34 J.s

03

Electron’s energy

The Energy and Time Uncertainty Principle,

∆t·∆E≥h2

Substituting values, and we get:

role="math" localid="1658386938701" ∆E≥1.05×10-342×1×10-9∆E≥5.3×10-26J

Therefore, the total energy is ∆E≥5.3×10-26J.

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Most popular questions from this chapter

Question: Atoms in a crystal form atomic planes at many different angles with respect to the surface. The accompanying figure shows the behaviors of representative incident and scattered waves in the Davisson-Germer experiment. A beam of electrons accelerated through 54 V is directed normally at a nickel surface, and strong reflection is detected only at an angle ϕof 500.Using the Bragg law, show that this implies a spacing D of nickel atoms on the surface in agreement with the known value of 0.22 nm.

A particle is connected to a spring and undergoes one-dimensional motion.

(a) Write an expression for the total (kinetic plus potential) energy of the particle in terms of its position x. its mass m, its momentum p, and the force constantof the spring.

(b) Now treat the particle as a wave. Assume that the product of the uncertainties in position and momentum is governed by an uncertainty relation∆p∆.r≈12h. Also assume that because xis on average. the uncertaintyis roughly equal to a typical value of|x|. Similarly, assume that∆p≈|p|. Eliminate pin favor of xin the energy expression.

(c) Find the minimum possible energy for the wave.

Question: A classmate studies Figures 12 and 17, then claims that when a spot appears, its location simultaneously establishes the particle’s -component of momentum, according to the angle from center, and its position (i.e., at the spot). How do you answer this claim?

A brag diffraction experiment is conducted using a beam of electrons accelerated through a1.0 kVpotential difference. (a) If the spacing between atomic planes in the crystal is0.1nm, at what angles with respect to the planes will diffraction maximum be observed? (b) If a beam ofX-rays products diffraction maxima at the same angles as the electron beam, what is theX-ray photon energy?

Question: Incandescent lightbulbs heat up a filament “white hot,” producing light of all wavelengths that has little to do with the filament’s composition. Gas vapor bulbs, such as sodium and mercury streetlights, produce colors that do depend on the gas in the bulb. Viewed with a diffraction grating (even a simple CD!), whereas the incandescent spectrum is continuous, that of a gas vapor (or fluorescent) bulb has characteristic lines. How is this indirect evidence of the wave nature of orbiting electrons?
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